Geometric Algebra: A natural representation of three-space James M. Chappell∗ and Derek Abbott School of Electrical and Electronic Engineering, University of Adelaide, SA 5005, Australia Azhar Iqbal a) School of Electrical and Electronic Engineering, University of Adelaide, SA 5005, Australia b) Centre for Advanced Mathematics and Physics, National University of Sciences & Technology, Sector H-12, Islamabad, Pakistan (Dated: April 17, 2013) Abstract Historically, there have been many attempts to produce the appropriate mathematical formalism for modeling the nature of physical space, such as Euclid's geometry, Descartes' system of Cartesian coordinates, the Argand plane, Hamilton's quaternions, and finally Gibbs' vector system using the dot and cross products. We conclude however, that Clifford's geometric algebra (GA), provides the most elegant description of space. Supporting this conclusion, we firstly show how geometric algebra encapsulates the key elements of the competing formalisms, such as complex numbers, quaternions and the dot and vector cross products and secondly we show how it provides an intuitive representation and manipulation of points, lines, areas and volumes. We also provide two key examples where GA has been found to provide an improved description of physical phenomena, electromagnetism and quantum theory. 1 I. INTRODUCTION Einstein once stated, `Everything should be made as simple as possible, but not one bit simpler', and in this paper we ask the question: `What is the simplest mathematical representation of three-dimensional physical space that is nevertheless complex enough to satisfactorily describe all its key properties?' The presence of five regular solids confirms the conclusion that we live in a three- dimensional world. If we lived in a world with four spatial dimensions, for example, we would be able to construct six regular solids, and in five dimensions and above we would find only three1. Also, the gravity and the electromagnetic force laws have been experimentally verified to follow an inverse square law to very high precision2, indicating the absence of ad- ditional macroscopic dimensions beyond three space dimensions. Hence a three-dimensional coordinate system, as proposed by Descartes, appears to be a suitable overall framework. For three-space, however, as well as positional coordinates, we also need to be able to repre- sent orientation or rotations at each point in this space. In the plane the algebra of complex numbers can be used for rotation and in three space, the algebra for rotations is given by Hamilton's quaternions. Hence, in order to form a unified algebra of three-space we need to integrate the complex numbers and quaternions within the framework of Cartesian coor- dinates. This was achieved by Clifford in 1873, who named his system, Geometric Algebra (GA). A. Historical development Around 547 BCE, opposing the world view of his time, Thales expressed his belief that every event on the earth had natural rather than mythological causes3. Pythagoras extended this notion of natural causation, by postulating that numbers and their relationships, underly all things. This mathematical approach by Pythagoras was masterfully applied by Euclid to geometry, deriving his famous set of geometrical theorems based on a few simple axioms, that formed the first comprehensive theory for the physical world. The next real breakthrough in mathematical science did not come though till the seventeenth century, and it has been extensively debated by historians, why there was such a slow down in the progress of science and mathematics following the Greek explosion. Various suggestions have been provided 2 to answer this, such as the Roman empire suppressing dissent and not sponsoring the arts, the ready availability of slaves obviating the need for work efficiencies4. However it has also been proposed that the algebraic and numerical system used by the Greeks, had inherent limitations, which were roadblocks to further progress5. For example the length along the p diagonal of a unit square, we know today as 2, being an irrational number, did not exist in the Greek numeric system, which was based solely on integers and their ratios. Another hindrance would have been the roman numerals which made numeric manipulation difficult. However with the arrival of Hindu-Arabic numbers in about 1000 AD into Europe, which included a zero that allowed positional representation for numbers, together with the accep- tance of negative numbers in 1545 AD, allowed for the concept of a complete number line to be developed. This then paved the way for Descartes to revolutionize the Greek system in 1637, by proposing a union of algebra and geometry using Cartesian coordinates. He stated `Just as arithmetic consists of only four or five operations, namely, addition, subtraction, multiplication, division and the extraction of roots, which may be considered a kind of di- vision, so in geometry, to find required lines it is merely necessary to add or subtract lines.' Descartes thus postulated an equivalence between line segments and numbers, something the Greeks were not prepared to do. This achievement is identified by John Stuart Mill, `the greatest single step ever made in the exact sciences'6. The Cartesian coordinate system proposed by Descartes, appears to become confused, however, with the later development of the Argand diagram, which, while isomorphic to the Cartesian plane, consists of one real and one imaginary axis, and so not rotationally symmetric. To add to the confusion, Hamilton in 1843 generalized the complex numbers to three space, defining the algebra of the quaternions using the basis elements i; j; k that can also be used as a substitute for three dimensional Cartesian coordinates. This confused state of affairs, on exactly how to represent three-space coordinates and rotations, was fi- nally resolved by William Clifford in 1873. Clifford adopted the Cartesian coordinate system of Descartes, but then also integrated the algebra of complex numbers and quaternions as the rotation operators within this space. Clifford also achieved a fulfillment of Descartes' original vision of a vector being able to be manipulated in the same way as normal num- bers, by deriving a multiplication and division operation for vectors that allowed them to be treated as algebraic variables. Additionally Clifford’s system extended this idea to its natural conclusion, allowing allowing not just lines, but also areas and volumes, and their 3 compositions, to be also treated in this same way. B. Clifford’s definition of three-space How did Clifford solve the problem of forming an integrated description of three-space combining Cartesian coordinates and the algebra of complex numbers and quaternions, as well as providing an algebraic treatment of lines, areas and volumes? Firstly, we represent the three degrees of freedom in a Cartesian coordinate system by the algebraic constants e1; e2 and e3 as shown in Fig. 1, which we define to have a positive square, 2 2 2 that is e1 = e2 = e3 = 1. The next crucial step is specify these elements as anticommuting, that is ejek = −ekej for j 6= k. These few definitions are sufficient to define Clifford’s system. Geometrically, the basis elements e1; e2; e3, the bivectors e2e3, e1e3 and e2e3 and the trivector e1e2e3, represent unit lines, unit areas, and unit volumes respectively. We also find that the compound algebraic elements, the bivectors e2e3, e1e3 and e2e3 all square to 2 minus one, for example, (e1e2) = e1e2e1e2 = −e1e1e2e2 = −1, using the anticommutivity and positive square of the basis elements. We can now identify an isomorphism of the three bivectors with the three quaternions of Hamilton, so that i $ e2e3, j $ e1e3, k $ e1e2. Also, p in the plane, the bivector e1e2 can be used as a replacement for the unit imaginary −1, forming a complex-like number a+ib, where we define i = e1e2. The final compound element, the trivector j = e1e2e3 also squares to minus one and commutes with all basis elements and p so is isomorphic to the scalar unit imaginary −1 in three dimensions. Now, because the unit imaginary is no longer required in Clifford’s system, and because the unit imaginary was first used in complex numbers that are isomorphic to GA in two dimensions, we will adopt the widely used symbol i = e1e2 to represent the bivector, and in three dimensions, we will adopt j = e1e2e3, a commonly used symbol in electrical engineering to represent the p unit imaginary. This distinction between two forms of the unit imaginary −1, as i and j in two and three dimensions respectively, has physical significance when describing Dirac's equation for the electron, described later. The basic elements of the algebra used to describe three dimensional space shown in Fig. 1. Now, using the trivector j we also find the relations e1e2 = je3, e3e1 = je2 and e2e3 = je1. 2 For example, je1 = e1e2e3e1 = e1e2e3 = e2e3, as required. These relations can be summarized by the relation eiej = jijkek, which we see describes the Pauli algebra, and hence we can use 4 FIG. 1: The basic elements of Clifford’s model for three space. This consists of three unit vectors e1; e2 and e3, three unit areas e2e3; e3e1 and e1e2 and a unit volume j = e1e2e3 Clifford’s basis vectors ek to replace the three Pauli matrices σk commonly used to describe quantum mechanical spin. The competing mathematical systems that Clifford unified is shown in Fig. 2. In the p plane the unit imaginary −1 is replaced with the bivector i = e12, using the subscript notation e12 = e1e2. Hamilton's three quaternions i; j and k representing rotations about the three available axes, can be replaced with the bivectors e23, e13 and e12 respectively, as shown, with the Cartesian axes described by unit vectors e1, e2 and e3. C. The Clifford vector product Using the three basis elements we can define a vector v = v1e1 +v2e2 +v3e3, where vi 2 <, and given a second vector u = u1e1 + u2e2 + u3e3, we can find their algebraic product using 5 FIG.